Classification of Vertex-Transitive Zonotopes

Abstract We give a full classification of vertex-transitive zonotopes. prove that zonotope is $$\Gamma $$ Γ -permutahedron for some finite reflection group \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ ⊂ O ( R d ) . The same holds true zonotopes in which all vertices are on common sphere, and edges the length. these then follows from groups. root systems can be characterized as those centrally symmetric sets vectors, intersections with half-spaces, contain exactly half congruent. provide further sufficient condition set to system.